About

Positional numeral systems encode quantity through digit placement relative to a radix r. A single misread digit in a hexadecimal memory address corrupts an entire data segment. Firmware engineers routinely convert between base-2, base-8, base-16, and base-10 under time pressure where manual errors cascade into hardware faults. This calculator performs exact arbitrary-precision integer arithmetic and conversion across all standard bases from 2 to 36, using digits 0 - 9 and letters A - Z. It validates every digit against the selected radix before computation, eliminating silent truncation bugs found in many online converters.

The tool handles numbers exceeding 2⁶⁴ via arbitrary-precision integer support. Arithmetic operations (addition, subtraction, multiplication, integer division, modulo, exponentiation) and bitwise operations (AND, OR, XOR, NOT, left/right shift) are computed exactly. Note: fractional base conversion uses fixed precision of 20 fractional digits and may exhibit rounding in non-terminating expansions (e.g., 0.1 in base 10 is non-terminating in base 2).

Formulas

A number N expressed in base r with digits d_k has the decimal value:

N = n∑k=0 d_k ⋅ r^k

where d_k ∈ {0, 1, …, r − 1} and k = 0 is the least significant digit.

To convert a decimal integer to base r, repeatedly divide by r and collect remainders in reverse order:

d_k = N mod r , N ← ⌊Nr⌋

For fractional parts, multiply by r and extract integer parts successively. This process may not terminate (e.g., 0.1₁₀ in base 2 is 0.0001100110011…).

Bitwise operations treat the number as a binary string. For two integers a and b:

a AND b → bit-by-bit logical conjunction
a OR b → bit-by-bit logical disjunction
a XOR b → bit-by-bit exclusive or
NOT a → bitwise complement
a << n ≡ a × 2ⁿ
a >> n ≡ ⌊a ÷ 2ⁿ⌋

where r = radix (base), d_k = digit at position k, N = numeric value, n = number of digit positions or shift amount.

Reference Data

Base	Name	Digits Used	Common Use	Example: 255₁₀
2	Binary	0-1	Digital circuits, CPU instructions	11111111
3	Ternary	0-2	Balanced ternary computers, information theory	100110
4	Quaternary	0-3	DNA nucleotide encoding	3333
5	Quinary	0-4	Tally systems, Bi-quinary coded decimal	2010
6	Senary	0-5	Dice notation, some natural languages	1103
7	Septenary	0-6	Days of the week, historical calendars	513
8	Octal	0-7	Unix file permissions, PDP-11 architecture	377
10	Decimal	0-9	Human counting, finance, science	255
12	Duodecimal	0-9, A - B	Time (12 hours), imperial units (dozen)	193
16	Hexadecimal	0-9, A - F	Memory addresses, color codes, MAC addresses	FF
20	Vigesimal	0-9, A - J	Maya numerals, French numbering (quatre-vingts)	CF
32	Base32	0-9, A - V	Crockford encoding, TOTP secret keys	7V
36	Base36	0-9, A - Z	URL shorteners, compact alphanumeric IDs	73
58	Base58 *	Alphanumeric minus 0OIl	Bitcoin addresses, IPFS hashes	Not supported (non-standard)
64	Base64 *	A - Z, a - z, 0-9, +/	Email encoding (MIME), data URIs	Not supported (encoding, not positional)

Frequently Asked Questions

The fraction 0.1₁₀ = 110, and 10 has a prime factor of 5 which is not a factor of 2. A fraction terminates in base r only when its denominator (in lowest terms) has no prime factors other than those of r. Since base 2 only contains the prime factor 2, any denominator with a factor of 5 produces a non-terminating binary expansion. This calculator truncates at 20 fractional digits.

JavaScript's Number type uses IEEE 754 double-precision floats, which lose integer precision above 2⁵³. This tool uses BigInt for all integer arithmetic, supporting arbitrary-precision integers limited only by available memory. Fractional parts use a scaled-integer approach: multiply the fractional digits by r²⁰, convert as BigInt, then reinsert the radix point.

For any base r from 11 to 36, digits 0 - 9 represent values 0-9, and letters A - Z represent values 10-35. Input is case-insensitive. For base 16, valid digits are 0 - 9 and A - F. Entering a digit whose value equals or exceeds the base (e.g., F in base 15) triggers a validation error.

For fixed-width integers (8, 16, 32, 64 bits), NOT flips every bit. For arbitrary-precision BigInt, JavaScript defines NOT(a) = −(a + 1), equivalent to two's complement with infinite sign extension. This calculator uses that definition. Positive inputs yield negative results and vice versa.

No. This tool converts mathematical numeric values between positional numeral systems. It does not display the internal IEEE 754 binary encoding (sign bit, exponent field, mantissa). For that purpose, use a dedicated IEEE 754 analyzer. The fractional conversion here shows the mathematical base-r expansion, not the hardware storage format.

Integer division truncates toward zero, matching the behavior of most programming languages (C, Java, Python's // operator). The remainder (modulo) operation is also provided. For example, 7 ÷ 2 = 3 with remainder 1. The tool displays both quotient and remainder when performing division.